Tailored Dispersion of Spectro‐Temporal Dynamics in Hot‐Carrier Plasmonics

Abstract Ultrafast optical switching in plasmonic platforms relies on the third‐order Kerr nonlinearity, which is tightly linked to the dynamics of hot carriers in nanostructured metals. Although extensively utilized, a fundamental understanding on the dependence of the switching dynamics upon optical resonances has often been overlooked. Here, all‐optical control of resonance bands in a hybrid photonic‐plasmonic crystal is employed as an empowering technique for probing the resonance‐dependent switching dynamics upon hot carrier formation. Differential optical transmission measurements reveal an enhanced switching performance near the anti‐crossing point arising from strong coupling between local and nonlocal resonance modes. Furthermore, entangled with hot‐carrier dynamics, the nonlinear correspondence between optical resonances and refractive index change results in tailorable dispersion of recovery speeds which can notably deviate from the characteristic lifetime of hot carriers. The comprehensive understanding provides new protocols for optically characterizing hot‐carrier dynamics and optimizing resonance‐based all‐optical switches for operations across the visible spectrum.

, where the step size for the AOI was decreased from 2 to 0.2 . Here we no longer see the jagged patterns indicating that such feature is simply an artifact of the lack of spectral and angular resolution of the simulation. This feature is observed due to the spectral linewidth of the resonance mode being much narrower compared to the magnitude of the spectral shift which can be seen in Figure S1b. The reflection and absorption maps are also plotted where we see a mismatch in the spectral and angular regions for maximum reflection and absorption.
The results are shown in Figure S1c and Figure

Distinguishing between lattice plasmon resonance and guided mode resonance
Both lattice plasmon (LP) mode and guided mode resonance (GMR) occur due to diffractive coupling and thus the two modes show similar spectral behavior, where the resonance locations are sensitive to the in-plane momentum of the impinging light. This makes it often confusing to properly identify whether a resonance mode is either a LP mode or a GMR under TM-polarized illumination. We deduce that the resonance dependent to the angle of incidence (AOI) that we observe under TM incidence is indeed an LP mode based on two reasons. First, the spectral locations of the high-Q resonance beyond 800 nm are well above the cut-off wavelength for TM waveguide mode in the current configuration. Second, the resonance dip at lower AOIs vanishes since LP mode is sensitive to the plasmonic response of the resonators and requires a strong out-of-plane dipole coupling. [1] Therefore, both the dampened plasmonic response at shorter wavelengths and small AOIs make the LP mode disappear, where the resonance should have been observed if the resonance originated from GMR. This also explains why we only see one resonance dip instead of two for the LP mode resonance at oblique incidence.

3. Parameters for the coupled oscillator model
The analytical expression of the eigenfrequencies of the two polariton modes that describes the experimental results most accurately was obtained by fitting Eq. (2) in the main text to the experimental values given in Figure 2d of the main text. The spectral location of the uncoupled LSPR mode was fitted as a constant, which its location lies within the spectrally broad resonance mode near 720 nm. The spectral location of the uncoupled SLR mode was fitted to the following equation: [2][3] (S1) Here, is the periodicity of the devised structure, is the refractive index of the surrounding environment of the plasmonic gratings, and is the angle of incidence. The spectral locations described in wavelengths were then converted into photon energy expressions.
Numerical fitting was executed via MATLAB where the fminsearch algorithm was implemented. function of the AOI. The situation however is more complicated than it seems due to the oblique incidence of the pump light relative to the probe light. Looking at Figure S2 where angle notations are graphically described, at the excitation condition with zero sample rotation the pump light has an angle of incidence ( ) of 10° in the y-z plane which can be understood as the azimuthal angle ( ) being 90°. As we increase the AOI ( ) for the probe light through sample rotation, both the AOI ( ) and azimuthal angle ( ) of the pump light changes. The relationship between the angles , , , and can be described through the following equations: The absorption of the nanograting structures at 840 nm can be simulated through COMSOL with the given angle of incidences and azimuthal angles. Given that the incident power level was kept constant, changing the AOI also modifies the intensity of the pump light due to varying cross sectional area of the laser beam where the intensity change is proportional to . Thus, the effective absorptance upon varying AOIs can be computed as: where is the pump power level at normal incidence. The numerical results of the absorbed power as a function of the sample rotation ( ) is shown in Figure S3.
Due to the initial angular offset, the polarization direction of the pump light relative to the grating also changes upon sample rotation. Following the coordinate conventions of Figure 1a in the main text, the polarization rotation of the TE and TM polarizations as a function of varying AOIs can be expressed as: Given that the angular offset of the pump light is 10°, the major portion of the incident power remains in the y-component of the electric field which is parallel to the grating structure regardless of the angle of incidence. Therefore, the variation in the excitation condition due to polarization rotation can be neglected.   Figure S4. A monotonic trend upon increasing probe intensity cannot be seen and therefore the influence of non-degenerate two-photon absorption to the transient response can be ruled out in our experimental results.

Figure S4
Transient response curves with varying probe intensities at . denotes the maximum probe intensity acheivable before the onset of saturation of the photodetector. The probing condition was fixed to TE polarization at normal incidence.

Simplified qualitative model for polariton modulation analysis
We first start with defining the linewidths and resonance wavelengths of the two uncoupled resonances at different AOIs. The linewidth for the LSP mode is retrieved at normal incidence, and the linewidth for the LP mode is retrieved at high AOIs where the perturbation due to coupling is small. We then compute the resonance locations and linewidths of the two polariton modes under all AOIs based on eq. (2) in the main text. The transmission spectra each resulting from the polaritons are assumed as Lorentzian curves, and the overall transmission spectra was obtained by averaging the contributions from the two polariton curves for simplicity.
We now repeat the process however with a perturbed linewidth for the LSP mode. Having obtained the angular dispersion of the perturbed and unperturbed transmission of the 1D crystal, the change in optical density can once again be calculated as , which gives the results we see in Figure 3e. Some exemplary curves visually describing the modelling process is given in Figure S5.
As discussed in the main text, Figure S6 shows that coupled oscillation modes formed from resonances with similar linewidths will not result in maximum modulation depths near the anti-crossing point. Here, we assigned linewidths of LP mode resonances to be the same as the LSP mode, and repeated the entire process described above. This further confirms our qualitative analysis, since there are no such trade-off relations between the static linewidth and the perturbed linewidth.
13  can be expressed as: As the reference for the estimation of refractive index change, we choose the transient response of our structure when probed by TE polarized probe at normal incidence. The linearized coefficients for induced relative permittivity change of gold (Au) as a function of effective electron temperature was retrieved from literature. [4] Given that major features of the transient response are located near 552nm, we neglect the contribution of the change in the intraband transition rate which is present in longer wavelengths. The modeled differential response is given in Figure S7a, and the subsequent refractive index change is plotted as a function of wavelength which is shown in Figure S7b. In the figures we see that the modeled differential response shows good agreement to our experimental results shown in Figure 4a in the main text, which implies that our estimation of the induced change in refractive index is indeed within a reasonable range compared to the actual value. Here, the discrepancy in the linear optical transmission predicted by the numerical simulation and the measured optical response is propagated throughout the differential response modelling, which can be observed as a slight blueshift of the modulation features numerically predicted compared to the experimental values.
Although beyond the scope of our work, the accuracy of the estimations can be further improved by incorporating the inhomogeneous three-temperature model (I3TM) and considering the contribution from the change in the intraband transition rate of Au for refractive index modelling. [5] Furthermore, instead of the linear approximation for the differential response, we can simulate the linear response of the structure with refractive indices at different time stamps.
This approach aligns better with our work given that the linearization assumes that the switching speed is identical throughout the entire wavelength and strictly follows the carrier relaxation time which is contrary to this study. In order to avoid such concerns, the influence of the coherent artifact was suppressed through a number of steps. First, a low pass filter and a moving average window was applied within the time window where the coherent artifact is existent to filter out the coherent artifact.
Second, the instrument response function (which has a much longer time constant compared to the oscillation cycles of the coherent artifact) limits the time constants to capture the ultrafast oscillation coming from the coherent artifact. Last, generally, three time constants (effectively two since one is set to infinity) used for the fitting cannot fully capture the coherent artifact given that it shows multiple oscillations. In order to have a complete picture on the dispersive nature of the recovery time, a detailed discussion on the temporal dynamics of the plasmonic crystal near the nodal points is required. We first start by looking at the transient response curve at one of the probing conditions that gives a fast recovery time, which is given in Figure S9. Here, we see a rapid oscillation in a short time window which occurs due to the coherent artifact, and then we see a transient response which the modulation flips in sign as time passes. As we see in Figure Figure 4 in the main text, the characteristic photon lifetime of resonance modes has often been pointed responsible for such effect. Larger characteristic photon lifetime of a resonance mode leads to increased time required to form steady-state optical resonance in the resonator, [6] which thus causes a delay in the recovery of the optical response upon perturbation. For the case of hot-carrier excitation, the photon lifetime delays re-formation of the static resonance response following the complete recovery of hotelectron induced refractive index change. However, the impact maybe subtle as the spectrally broad nature of the resonance modes in our work leads to a relatively short photon lifetime, implying that this isn't the main factor of the slower modulation speed near resonance wavelengths. We take an GMR mode at an AOI of 16° for example to estimate the timescale of the longest possible photon lifetime for the resonance modes supported by our sample. The photon lifetime can be obtained through the Q-factor of the resonance modes, which is calculated by . As we see in Figure S9, is the resonance wavelength which is located at 656 nm, and is the full width at half maximum (FWHM) which is approximately 22nm.
Therefore, we can estimate the Q-factor to be ~30 where and thus the photon lifetime is roughly ~10 fs which is much shorter than the timescale for the recovery speed. The photon lifetime was calculated by where is the angular resonance frequency.